Regularity and stability results in variational problems

阿莱西奥·菲加利的这项数学研究项目集中在变分和偏微分方程中的几个问题上。这些问题包括最优运输问题、泛函不等式的稳定性问题和Mumford-Shah泛函。最优运输问题包括找到最便宜的方法将质量分布从一个地方运送到另一个地方。除了是变分中的一个自然问题外，它还与偏微分方程、黎曼几何和概率论有关。泛函不等式的稳定性问题包括了解某一不等式的极小值是否在某种适当意义下是稳定的。为了理解和/或预测物理现象的时间演化，这是一个重要的问题。例如，定量稳定性结果被用来量化给定物理系统向其稳定状态的收敛速度，它们也可以用来理解系统在外部因素(例如，外力)影响下的变化程度。Mumford-Shah泛函是图像分割中的经典模型，用于从模糊图像中提取有意义的不连续(对应于对象的边缘、阴影和重叠对象)。Mumford-Shah泛函极小子的正则性还远未被理解，而了解界面的光滑性及其拓扑性质是一个重要而具有挑战性的问题。Alessio Figalli在这个数学研究项目中所研究的问题在其他科学领域都有重要的应用。例如，最优运输问题是经济学中的一个基本问题，它进一步应用于气象学、生物学和人口动力学；本项目中研究的Mumford-Shah泛函在图像处理中有应用(它允许从模糊的图像中提取出好的图像)。该项目中的部分问题将用于本科生、研究生和博士后研究员的培养。菲加利的几名博士生和博士后研究员将从事这些领域的研究，并将通过发表研究论文和课堂讲稿以及开发课程和研讨会来广泛传播所取得的成果。

This mathematics research project by Alessio Figalli is focused on several problems in the calculus of variations and partial differential equations. These include the optimal transport problem, the issue of stability in functional inequalities, and the Mumford-Shah functional. The optimal transport problem consists of finding the least expensive way to transport a distribution of mass from one place to another. In addition to being a natural problem in the calculus of variations, it is also related to partial differential equations, Riemannian geometry, and probability. The issue of stability in functional inequalities consists of understanding whether a minimizer of some inequality is stable in some suitable sense. This is an important issue in order to understand and/or predict the evolution in time of a physical phenomenon. For instance, quantitative stability results are used to quantify the rate of convergence of a given physical system to its steady state, and they can also be used to understand the extent to which the system changes under the influence of external factors (for instance, external forces). The Mumford-Shah functional is a classical model in image segmentation which is used to extract from a blurry image the meaningful discontinuities (which correspond to edges of objects, shadows, and overlapping objects). The regularity properties of minimizers of the Mumford-Shah functional are still far from being understood, and understanding the smoothness of the interfaces and their topological properties is an important and challenging problem.All the problems investigated in this mathematics research project by Alessio Figalli have important applications in other areas of sciences. For instance, the optimal transport problem is a fundamental problem in economics, with further applications to meteorology, biology, and population dynamics; the Mumford-Shah functional, which is studied in this project, has applications to image processing (it allows to extract good images out of blurry ones). Some of the problems in this project will be used in the training of undergraduate students, graduate students and postdoctoral fellows. Several of Figalli's PhD students and postdoctoral fellows will engage in research in these areas, and the results obtained will be widely disseminated via the publication of research papers and lecture notes, as well as through the development of courses and seminars.